How can my animation animate if there is no vertex weights?
Your screenshot is showing the weights (no weights) relatively to one bone/vertex group only.

Every vertex can be weighted relatively to every bone of the rig (and even to multiple rigs). So, you must check the weight painting of every vertex group of the mesh to see which are the weights. In alternative you can select one vertex and read which are its weights in the "N" properties panel, vertex weights tab.